Converting from Other Bases to Base 10 Using the Expanded Notation

Worked Example 1.3.1:

Convert the following numbers to base 10 using expanded notation:

a. 107_eight
b. 1211_three
c. 203_six 
d. 4102_five
e. 13011_four

Solution

a. 107_eight

The table shows the place values for 107. Remember that we count 0 from the last digit from right to left. We multiply each digit by the base (which is 8 for this particular question) raised to the power of the place value.

107_eight = 1 × 8² + 0 × 8¹ + 7 × 8^o

= 64 + 0 + 7

= 71_ten

∴ 107_eight = 71_ten

b. 1211_three

1211_three = 1 × 3³ + 2 × 3² + 1 × 3¹ + 1 × 3^o

= 27 + 18 + 3 + 1

=  49_ten

∴ 1211_three = 49_ten

c. 203_six 

203_six  = 2 × 6² + 0 × 6¹ + 3 × 6^o

= 72 + 0 + 3

=  75_ten

∴ 203_six =  75_ten

d. 4102_five

4102_five = 4 × 5³ + 1 × 5² + 0 × 5¹ + 2 × 5^o

= 500 + 25 + 0 + 2

=  527_ten

∴ 4102_five =  527_ten

e. 13011_four

13011_four = 1 × 4⁴ + 3 × 4³ + 0 × 4² + 1 × 4¹ + 1 × 4^o

= 256 + 192 + 0 + 4 + 1

=  453_ten

∴ 13011_four =  453_ten